mit phd international trade 6: ricardo-viner and heckscher
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14.581 MIT PhD International Trade – Lecture 6: Ricardo-Viner and Heckscher-Ohlin
Models (Theory II) –
Dave Donaldson
Spring 2011
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Today’s Plan
Two-by-two-by-two Heckscher-Ohlin model
Integrated equilibrium
Heckscher-Ohlin Theorem
High-dimensional issues
Classical theorems revisited
Heckscher-Ohlin-Vanek Theorem
Assignment models (briefiy)
Two-by-two-by-two Heckscher-Ohlin model Basic environment
• Results derived in previous lecture hold for small open economies.
• Relative good prices were taken as exogenously given.
• We now turn world economy with two countries, North and South.
• We maintain the two-by-two HO assumptions:
• There are two goods, g = 1,2, and two factors, k and l . • Identical technology around the world, yg = fg (kg , lg ). Identical homothetic preferences around the world, dc = αg (p)I c .• g
• Question What is the pattern of trade in this environment?
“ ’ ”
Two-by-two-by-two Heckscher-Ohlin model Strategy ( Samuelson s Angel )
• Start from Integrated Equilibrium ≡ competitive equilibrium that would prevail if both goods and factors were freely traded.
• Consider Free Trade Equilibrium ≡ competitive equilibrium that prevails if goods are freely traded, but factors are not.
• Ask: Can free trade equilibrium reproduce integrated equilibrium?
• Answer turns out to be yes, if factor prices are equalized through trade.
• In this situation, one can then use homotheticity to go from differences in factor endowments to the pattern of trade
� �
� �
Two-by-two-by-two Heckscher-Ohlin model Integrated equilibrium
• Integrated equilibrium corresponds to (p, ω, y ) such that:
(ZP) : p = A� (ω) ω (1)
(GM) : y = α (p) ω�v (2)
(FM) : v = A (ω) y (3)
where:
• p ≡ (p1, p2), ω ≡ (w , r ), A (ω) ≡ afg (ω) , y ≡ (y1, y2) is the vector of total world output, v ≡ (l , k) is the vector of total world endowments, and α (p) ≡ [α1 (p) , α2 (p)] .
• A (ω) derives from cost-minimization.
• α (p) derives from utility-maximization.
• So this is the equilibrium of the world economy if factors were allowed to be mobile.
Two-by-two-by-two Heckscher-Ohlin model Free trade equilibrium
• Free trade equilibrium corresponds to (pt , ωn , ωs , yn , ys ) such that:
(ZP) : pt ≤ A� (ωc )� ωc � � for c = n, s � (4)
(GM) : yn + ys = α pt ωn�vn + ωs �vs (5)
(FM) : vc = A (ωc ) yc for c = n, s (6)
where (4) holds with equality if good is produced in country c .
• Definition: Free trade equilibrium replicates integrated equilibrium if ∃ (yn , ys ) ≥ 0 such that (p, ω, ω, yn , ys ) satisfy conditions (4)-(6)
Two-by-two-by-two Heckscher-Ohlin model Factor Price Equalization (FPE) Set
• Definition (vn , vs ) are in the FPE set if ∃ (yn , ys ) ≥ 0 such that condition (6) holds for ωn = ωs = ω.
• Lemma If (vn , vs ) is in the FPE set, then the free trade equilibrium replicates the integrated equilibrium
• Proof: By definition of the FPE set, ∃ (yn , ys ) ≥ 0 such that
vc = A (ω) yc .
So Condition (6) holds. Since v = vn + vs , this implies
v = A (ω) (yn + ys ) .
Combining this expression with condition (3), we obtain yn + ys = y . Since ωn�vn + ωs �vs = ω�v , Condition (5) holds as well. Finally,Condition (1) directly implies (4) holds.
Two-by-two-by-two Heckscher-Ohlin model Integrated equilibrium: graphical analysis
• Factor market clearing in the integrated equilibrium:
a1(ω)
k
O
l
a2(ω)
y2a2(ω)
y1a1(ω)
v
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Two-by-two-by-two Heckscher-Ohlin model The Parallelogram
• FPE set ≡ (vn , vs ) inside the parallelogram
vs
vn
ks
ls
a1(ω)
kn
On
ln
a2(ω)
y2a2(ω)
y1a1(ω)
v
Os
• When vn and vs are inside the parallelogram, we say that they belong to the same diversification cone.
• This is a very different way of approaching FPE than FPE Theorem. • Here, we have shown that there can be FPE iff factor endowments are not too dissimilar, whether or not there are no FIR.
• Instead of taking prices as given– whether or not they are consistent with integrated equilibrium– we take factor endowments as primitives.
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Two-by-two-by-two Heckscher-Ohlin model Heckscher Ohlin Theorem: graphical analysis
• Suppose that (vn , vs ) is in the FPE set.
• HO Theorem In the free trade equilibrium, each country will export the good that uses its abundant factor intensively.
Slope = w/r
C
vs
vn
ks
ls
kn
On
ln
v
Os
• Outside the FPE set, additional technological and demand considerations matter (e.g. FIR or no FIR).
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Two-by-two-by-two Heckscher-Ohlin model Heckscher Ohlin Theorem: alternative proof
• The HO Theorem can also be derived using the Rybczynski effect:
Rybczynski theorem y2 n /y1
n > y2 s /y1
s for any p.⇒
Homotheticity c2 n /cn = c2
s /c1 s for any p.⇒ 1
This implies p2 n /p1
n < p2 s /p1
s under autarky.
Law of comparative advantage HO Theorem. ⇒
Two-by-two-by-two Heckscher-Ohlin model Trade and inequality
Predictions of HO and SS Theorems are often combined: • HO Theorem p2
n /p1 n < p2/p1 < p2
s /p1s .• ⇒
SS Theorem Moving from autarky to free trade, real return of • ⇒abundant factor increases, whereas real return of scarce factor decreases.
• If North is skill-abundant relative to South, inequality increases in the North and decreases in the South.
• So why may we observe a rise in inequality in the South in practice? Perhaps:
• Southern countries are not moving from autarky to free trade.
• Technology is not identical around the world.
Preferences are not homothetic and identical around the world. •
• There are more than two goods and two countries in the world.
Two-by-two-by-two Heckscher-Ohlin model Trade volumes
• Let us define trade volumes as the sum of exports plus imports.
• Inside FPE set, iso-volume lines are parallel to diagonal (HKa p.23).
• The further away from the diagonal, the larger the trade volumes. • Factor abundance rather than country size determines trade volume.
ks
ls
a1(ω)
kn
On
ln
a2(ω)
y2a2(ω)
y1a1(ω)
Os
• If country size affects trade volumes in practice, what should we infer?
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Today’s Plan
Two-by-two-by-two Heckscher-Ohlin model
Integrated equilibrium
Heckscher-Ohlin Theorem
High-dimensional issues
Classical theorems revisited
Heckscher-Ohlin-Vanek Theorem
Assignment models (briefiy)
High-Dimensional Predictions FPE (I): More factors than goods
• Suppose now that there are F factors and G goods. • By definition, (vn , vs ) is in the FPE set if ∃ (yn , ys ) ≥ 0 s.t. vc = A (ω) yc for c = n, s.
• If F = G (“even case”), the situation is qualitatively similar. • If F > G , the FPE set will be “measure zero”: {v |v = A (ω) yc for yc ≥ 0} is a G -dimensional cone in F -dimensional space.
• Example: “Standard Macro” model with 1 good and 2 factors.
vs
vn
ks
ls
a(ω)
kn
On
ln
Os
High-Dimensional Predictions FPE (II): More goods than factors
• If F < G , there will be indeterminacies in production, (yn , ys ), and so, trade patterns, but FPE set will still have positive measure.
• Example: 3 goods and 2 factors:
y1a1(ω)
y2a2(ω)
y3a3(ω)
a2(ω)
vs
vn
ks
ls
a1(ω)
kn
On
ln
a3(ω)
v
Os
• By the way, are there more goods than factors in the world?
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High-Dimensional Predictions Stolper Samuelson type results (I): Friends and Enemies
• SS Theorem was derived by differentiating the zero-profit conditions.
• With an arbitrary number of goods and factors, we still have
�pg = ∑f θfg w�f , (7)
where wf is the price of factor f and θfg ≡ wf afg (ω) /cg (ω) . • Now suppose that p�g0 > 0, whereas p�g = 0 for all g �= g0.
• Equation (7) immediately implies the existence of f1 and f2 s.t.
w�f1 �pg0 > p�g = 0 for all g = g0,≥ �w�f2 < p�g = 0 < p�g0 for all g = g0.
• So every good is “friend” to some factor and “enemy” to some other (Jones and Scheinkman 1977)
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� � � � � � � �
High-Dimensional Predictions Stolper Samuelson type results (II): Correlations
• Ethier (1984) also provides the following variation of SS Theorem.
• If good prices change from p to p�, then the associated change in factor prices, ω� − ω, must satisfy
ω� − ω A (ω0 ) p� − p > 0, for some ω0 between ω and ω�.
Proof: • Define f (ω) = ωA (ω) (p� − p). Mean value theorem implies
f ω� = ωA (ω) p� − p + ω� − ω [A (ω0 ) + ω0dA (ω0)] p� − p
for some ω0 between ω and ω�. Cost-minimization at ω0 requires
ω0dA (ω0 ) = 0.
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� � � � � �
� � � �
High-Dimensional Predictions Stolper Samuelson type results (II): Correlations
• Proof (Cont.): Combining the two previous expressions, we obtain
f ω� − f (ω) = ω� − ω A (ω0) p� − p .
From the zero profit conditions, we know that p = ωA (ω) and p� = ω�A (ω�). Thus � � � � � �
f ω� − f (ω) = p� − p p� − p > 0.
The last two expressions imply
ω� − ω A (ω0) p� − p > 0.
• Interpretation: Tendency for changes in good prices to be accompanied by raises in prices of factors used intensively in goods whose prices have gone up.
• But what is ω0?
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High-Dimensional Predictions Rybczynski type results
• Rybczynski Theorem was derived by differentiating the factor market clearing conditions.
• If G = F > 2, same logic implies that increase in endowment of one factor decreases output of one good and increases output of another (Jones and Scheinkman 1977).
• If G < F , increase in endowment of one factor may increase output of all goods (Ricardo-Viner).
• In this case, we still have the following correlation (Ethier 1984) � � � � � � � � v � − v A (ω0) y � − y = v � − v v � − v > 0.
• If G > F , inderteminacies in production imply that we cannot predict changes in output vectors.
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High-Dimensional Predictions Heckscher Ohlin type results
• Since HO Theorem derives from Rybczynski effect + homotheticity, problems of generalization in the case G < F and F > G carry over to the Heckscher-Ohlin Theorem.
• If G = F > 2, we can invert the factor market clearing condition
c cy = A−1 (ω) v .
• By homotheticity, the vector of consumption in country c satisfies
dc = scd
where sc ≡ c’s share of world income, and d ≡ world consumption.• Good and factor market clearing requires
d = y = A−1 (ω) v .
• Combining the previous expressions, we get net exports
tc c c c≡ y − dc = A−1 (ω) (v − s v ) .
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High-Dimensional Predictions Heckscher Ohlin Vanek Theorem
• Without assuming that G = F , we can derive sharp predictions if we focus on the G � F case and on the factor content of trade rather than commodity trade.
• We define the net exports of factor f by country c as
τcf = ∑g afg (ω) tgc .
• In matrix terms, this can be rearranged as
τc = A (ω) tc .
• HOV Theorem In any country c, net exports of factors satisfy
τc c c = v − s v .
• So countries should export the factors in which they are abundant compared to the world: vf
c > sc vf .
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Today’s Plan
Two-by-two-by-two Heckscher-Ohlin model
Integrated equilibrium
Heckscher-Ohlin Theorem
High-dimensional issues
Classical theorems revisited
Heckscher-Ohlin-Vanek Theorem
Assignment models (briefiy)
Assignment Models Basic Idea
• With 2 goods and 2 factors, neoclassical trade models lead to sharp comparative static predictions.
• With more than 2 goods and 2 factors, however, their predictions become weak and unintuitive.
• “Assignment approach” consists in imposing more structure on technology in order to transform analysis into an assignment problem (which has more success in high dimensions).
• Main assumption: Constant marginal product for all factors (as in a Ricardian model).
Main benefit: • Side-step many mathematical diffi culties to derive strong and intuitive predictions in high-dimensional environments.
Example: Costinot and Vogel (2009) Factor endowments
• Consider a world economy with two countries, Home and Foreign.
• There is a continuum of goods with skill-intensity σ ∈ Σ ≡ [σ, σ] .
• There is a continuum of workers with skill s ∈ S ≡ [s, s ] .
• V (s),V ∗(s) > 0 is the inelastic supply of workers with skill s.
• Home is skill-abundant relative to Foreign:
V (s �) V (s �) V (s)
> V (s)
for any s � ≥ s.
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Example: Costinot and Vogel (2009) Technology and preferences
• Technology is the same around the world.
• Workers are perfect substitutes in the production of each task:
Y (σ) = A (s, σ) L (s, σ) ds. s ∈S
• A (s, σ) > 0 is strictly log-supermodular:
A (s, σ) > A (s �, σ)
, for all s > s � and σ > σ�.A (s, σ�) A (s �, σ�)
• Consumers have identical CES preferences around the world: �� � ε
εU = [C (σ)] ε−1 dσ
ε−1 .
σ∈Σ
Example: Costinot and Vogel (2009) Results
• Generalizations of all two-by-two results: FPE, Stolper-Samuelson, Rybczynski, Heckscher-Ohlin.
• More importantly, model can be used to look at new phenomena.
• Example: North-North trade
V (s �) V ∗(s �) V (s)
> V ∗(s)
for any s � ≥ s ≥ s�, V (s �) V ∗(s �) V (s)
< V ∗(s)
for any �s ≥ s � ≥ s.
• One can show that trade integration leads to wage polarization in the more “diverse” country and wage convergence in the other country.
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14.581 International Economics I Spring 2011
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